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Volatility Given Delta

The mapping $\sigma\mapsto\Delta=\phi e^{-r_f\tau}{\cal N}(\phi d_+)$ is not one-to-one. Thus using just the delta to retrieve the volatility of an option is not advisable. The two solutions are given by

\begin{displaymath} \sigma_{\pm}=\frac{1}{\sqrt{\tau}}\left\{\phi{\cal N}^{-1}(\... ...phi\Delta e^{r_f\tau}))^2-\sigma\sqrt{\tau}(d_++d_-)}\right\}. \end{displaymath} (51)



2000-06-11
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